A Mind For Numbers: How to Excel at Math and Science (Even if You Flunked Algebra)

Whether you are a student struggling to fulfill a math or science requirement, or you are embarking on a career change that requires a higher level of math competency, A Mind for Numbers offers the tools you need to get a better grasp of that intimidating but inescapable field. Engineering professor Barbara Oakley knows firsthand how it feels to struggle with math. She flunked her way through high school math and science courses, before enlisting in the army immediately after graduation. When she saw how her lack of mathematical and technical savvy severely limited her optionsboth to rise in the military and to explore other careersshe returned to school with a newfound determination to re-tool her brain to master the very subjects that had given her so much trouble throughout her entire life.

In A Mind for Numbers, Dr. Oakley lets us in on the secrets to effectively learning math and sciencesecrets that even dedicated and successful students wish they’d known earlier. Contrary to popular belief, math requires creative, as well as analytical, thinking. Most people think that there’s only one way to do a problem, when in actuality, there are often a number of different solutionsyou just need the creativity to see them. For example, there are more than three hundred different known proofs of the Pythagorean Theorem. In short, studying a problem in a laser-focused way until you reach a solution is not an effective way to learn math. Rather, it involves taking the time to step away from a problem and allow the more relaxed and creative part of the brain to take over. A Mind for Numbers shows us that we all have what it takes to excel in math, and learning it is not as painful as some might think!

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contents
PRAISE FOR A MIND FOR NUMBERS
TITLE PAGE
COPYRIGHT
DEDICATION
EPIGRAPH
FOREWORD by Terrence J. Sejnowski, Francis Crick Professor, Salk Institute for Biological Studies
PREFACE by Jeffrey D. Karpicke, James V. Bradley Associate Professor of Psychological Sciences, Purdue University
NOTE TO THE READER
1 Open the Door
2 Easy Does It:
Why Trying Too Hard Can Sometimes Be Part of the Problem
3 Learning Is Creating:
Lessons from Thomas Edison’s Frying Pan
4 Chunking and Avoiding Illusions of Competence:
The Keys to Becoming an “Equation Whisperer”
5 Preventing Procrastination:
Enlisting Your Habits (“Zombies”) as Helpers
6 Zombies Everywhere:
Digging Deeper to Understand the Habit of Procrastination
7 Chunking versus Choking:
How to Increase Your Expertise and Reduce Anxiety
8 Tools, Tips, and Tricks
9 Procrastination Zombie Wrap-Up
10 Enhancing Your Memory
11 More Memory Tips
12 Learning to Appreciate Your Talent
13 Sculpting Your Brain
14 Developing the Mind’s Eye through Equation Poems
15 Renaissance Learning
16 Avoiding Overconfidence:
The Power of Teamwork
17 Test Taking
18 Unlock Your Potential
AFTERWORD BY DAVID B. DANIEL, PH.D., PROFESSOR, PSYCHOLOGY DEPARTMENT, JAMES MADISON UNIVERSITY
ACKNOWLEDGMENTS
ENDNOTES
REFERENCES
CREDITS
INDEX
{ 14 }
developing the mind’s eye through equation poems
Learn to Write an Equation Poem—Unfolding Lines That Provide a Sense of What Lies Beneath a Standard Equation
Poet Sylvia Plath once wrote: “The day I went into physics class it was death.”1 She continued:
A short dark man with a high, lisping voice, named Mr. Manzi, stood in front of the class in a tight blue suit holding a little wooden ball. He put the ball on a steep grooved slide and let it run down to the bottom. Then he started talking about let a equal acceleration and let t equal time and suddenly he was scribbling letters and numbers and equals signs all over the blackboard and my mind went dead.
Mr. Manzi had, at least in this semiautobiographical retelling of Plath’s life, written a four-hundred-page book with no drawings or photographs, only diagrams and formulas. An equivalent would be trying to appreciate Plath’s poetry by being told about it, rather than being able to read it for yourself. Plath was, in her version of the story, the only student to get an A, but she was left with a dread for physics.
“What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?”
—David Eugene Smith, American mathematician and educator
Physicist Richard Feynman’s introductory physics classes were entirely different. Feynman, a Nobel Prize winner, was an exuberant guy who played the bongos for fun and talked more like a down-to-earth taxi driver than a pointy-headed intellectual.
When Feynman was about eleven years old, an off-the-cuff remark had a transformative impact on him. He remarked to a friend that thinking is nothing more than talking to yourself inside.
“Oh yeah?” said Feynman’s friend. “Do you know the crazy shape of the crankshaft in a car?”
“Yeah, what of it?”
“Good. Now tell me: How did you describe it when you were talking to yourself?”
It was then that Feynman realized that thoughts can be visual as well as verbal.2
He later wrote about how, when he was a student, he had struggled to imagine and visualize concepts such as electromagnetic waves, the invisible streams of energy that carry everything from sunlight to cell phone signals. He had difficulty describing what he saw in his mind’s eye.3 If even one of the world’s greatest physicists had trouble imagining how to see some (admittedly difficult-to-imagine) physical concepts, where does that leave us normal folks?
We can find encouragement and inspiration in the realm of poetry.4 Let’s take a few poetic lines from a song by American singer-songwriter Jonathan Coulton, called “Mandelbrot Set,”5 about a famous mathematician, Benoit Mandelbrot.
Mandelbrot’s in heaven
He gave us order out of chaos, he gave us hope where there was none
His geometry succeeds where others fail
So if you ever lose your way, a butterfly will flap its wings
From a million miles away, a little miracle will come to take you home
The essence of Mandelbrot’s extraordinary mathematics is captured in Coulton’s emotionally resonant phrases, which form images that we can see in our own mind’s eye—the gentle flap of a butterfly’s wings that spreads and has effects even a million miles away.
Mandelbrot’s work in creating a new geometry allowed us to understand that sometimes, things that look rough and messy—like clouds and shorelines—have a degree of order to them. Visual complexity can be created from simple rules, as evidenced in modern animated movie-making magic. Coulton’s poetry also alludes to the idea, embedded in Mandelbrot’s work, that tiny, subtle shifts in one part of the universe ultimately affect everything else.
The more you examine Coulton’s words, the more ways you can see it applied to various aspects of life—these meanings become clearer the more you know and understand Mandelbrot’s work.
There are hidden meanings in equations, just as there are in poetry. If you are a novice looking at an equation in physics, and you’re not taught how to see the life underlying the symbols, the lines will look dead to you. It is when you begin to learn and supply the hidden text that the meaning slips, slides, then finally leaps to life.
In a classic paper, physicist Jeffrey Prentis compares how a brand-new student of physics and a mature physicist look at equations.6 The equation is seen by the novice as just one more thing to memorize in a vast collection of unrelated equations. More advanced students and physicists, however, see with their mind’s eye the meaning beneath the equation, including how it fits into the big picture, and even a sense of how the parts of the equation feel.
“A mathematician who is not at the same time something of a poet will never be a full mathematician.”
—German mathematician Karl Weierstrass
When you see the letter a, for acceleration, you might feel a sense of pressing on the accelerator in a car. Zounds! Feel the car’s acceleration pressing you back against the seat.
Do you need to bring these feelings to mind every time you look at the letter a? Of course not; you don’t want to drive yourself crazy remembering every little detail underlying your learning. But that sense of pressing acceleration should hover as a chunk in the back of your mind, ready to slip into working memory if you’re trying to analyze the meaning of a when you see it roaming around in an equation.
Similarly, when you see m, for mass, you might feel the inertial laziness of a fifty-pound boulder—it takes a lot to get it moving. When you see the letter f, for force, you might see with your mind’s eye what lies underneath force—that it depends on both mass and acceleration: m·a, as in the equation f = m·a. Perhaps you can feel what’s behind the f as well. Force has built into it a heaving oomph (acceleration), against the lazy mass of the boulder.
Let’s build on that just a wee bit more. The term work in physics means energy. We do work (that is, we supply energy) when we push (force) something through a distance. We can encrypt that with poetic simplicity: w = f×d. Once we see w for work, then we can imagine with our mind’s eye, and even our body’s feelings, what’s behind it. Ultimately, we can distill a line of equation poetry that looks like this:
w
w = f·d
w = (ma)·d
Symbols and equations, in other words, have a hidden text that lies beneath them—a meaning that becomes clear once you are more familiar with the ideas. Although they may not phrase it this way, scientists often see equations as a form of poetry, a shorthand way to symbolize what they are trying to see and understand. Observant people recognize the depth of a piece of poetry—it can have many possible meanings. In just the same way, maturing students gradually learn to see the hidden meaning of an equation with their mind’s eye and even to intuit different interpretations. It’s no surprise to learn that graphs, tables, and other visuals also contain hidden meaning—meaning that can be even more richly represented in the mind’s eye than on the page.
Simplify and Personalize Whatever You Are Studying
We’ve alluded to this before, but it’s worth revisiting now that we’ve got better insight into how to imagine the ideas that underlie equations. One of the most important things we can do when we are trying to learn math and science is to bring the abstract ideas to life in our minds. Santiago Ramón y Cajal, for example, treated the microscopic scenes before him as if they were inhabited by living creatures that hoped and dreamed just as people themselves do.7 Cajal’s colleague and friend, Sir Charles Sherrington, who coined the word synapse, told friends that he had never met another scientist who had this intense ability to breathe life into his work. Sherrington wondered whether this might have been a key contributing factor to Cajal’s level of success.
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Einstein was able to imagine himself as a photon.8 We can gain a sense of what Einstein saw by looking at this beautiful vision by Italian physicist Marco Bellini of an intense laser pulse (the one in front), being used to measure the shape of a single photon (the one in the back).
Einstein’s theories of relativity arose not from his mathematical skills (he often needed to collaborate with mathematicians to make progress) but from his ability to pretend. He imagined himself as a photon moving at the speed of light, then imagined how a second photon might perceive him. What would that second photon see and feel?
Barbara McClintock, who won the Nobel Prize for her discovery of genetic transposition (“jumping genes” that can change their place on the DNA strand), wrote about how she imagined the corn plants she studied: “I even was able to see the internal parts of the chromosomes—actually everything was there. It surprised me because I actually felt as if I were right down there and these were my friends.”9
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Pioneering geneticist Barbara McClintock imagined gigantic versions of the molecular elements she was dealing with. Like other Nobel Prize winners, she personalized—even made friends with—the elements she was studying.
It may seem silly to stage a play in your mind’s eye and imagine the elements and mechanisms you are studying as living creatures, with their own feelings and thoughts. But it is a method that works—it brings them to life and helps you see and understand phenomena that you couldn’t intuit when looking at dry numbers and formulas.
Simplifying is also important. Richard Feynman, the bongo-playing physicist we met earlier in this chapter, was famous for asking scientists and mathematicians to explain their ideas in a simple way so that he could grasp them. Surprisingly, simple explanations are possible for almost any concept, no matter how complex. When you cultivate simple explanations by breaking down complicated material to its key elements, the result is that you have a deeper understanding of the material.10 Learning expert Scott Young has developed this idea in what he calls the Feynman technique, which asks people to find a simple metaphor or analogy to help them grasp the essence of an idea.11
The legendary Charles Darwin would do much the same thing. When trying to explain a concept, he imagined someone had just walked into his study. He would put his pen down and try to explain the idea in the simplest terms. That helped him figure out how he would describe the concept in print. Along those lines, the website Reddit.com has a section called “Explain like I’m 5” where anyone can make a post asking for a simple explanation of a complex topic.12
You may think you really have to understand something in order to explain it. But observe what happens when you are talking to other people about what you are studying. You’ll be surprised to see how often understanding arises as a consequence of attempts to explain to others and yourself, rather than the explanation arising out of your previous understanding. This is why teachers often say that the first time they ever really understood the material was when they had to teach it.
IT’S NICE TO GET TO KNOW YOU!
“Learning organic chemistry is not any more challenging than getting to know some new characters. The elements each have their own unique personalities. The more you understand those personalities, the more you will be able to read their situations and predict the outcomes of reactions.”
—Kathleen Nolta, Ph.D., Senior Lecturer in Chemistry and recipient of the Golden Apple Award, recognizing excellence in teaching at the University of Michigan
NOW YOU TRY!
Stage a Mental Play
Imagine yourself within the realm of something you are studying—looking at the world from the perspective of the cell, or the electron, or even a mathematical concept. Try staging a mental play with your new friends, imagining how they feel and react.
Transfer—Applying What You’ve Learned in New Contexts
Transfer is the ability to take what you learn in one context and apply it to something else. For example, you may learn one foreign language and then find that you can pick up a second foreign language more easily than the first. That’s because when you learned the first foreign language, you also acquired general language-learning skills, and potentially similar new words and grammatical structures, that transferred to your learning the second foreign language.13
Learning math by applying it only to problems within a specific discipline, such as accounting, engineering, or economics, can be a little like deciding that you are not really going to learn a foreign language after all—you’re just going to stick to one language and just learn a few extra English vocabulary words. Many mathematicians feel that learning math through entirely discipline-specific approaches makes it more difficult for you to use mathematics in a flexible and creative way.
Mathematicians feel that if you learn math the way they teach it, which centers on the abstract, chunked essence without a specific application in mind, you’ve captured skills that are easy for you to transfer to a variety of applications. In other words, you’ll have acquired the equivalent of general language-learning skills. You may be a physics student, for example, but you could use your knowledge of abstract math to quickly grasp how some of that math could apply to very different biological, financial, or even psychological processes.
This is part of why mathematicians like to teach math in an abstract way, without necessarily zooming in on applications. They want you to see the essence of the ideas, which they feel makes it easier to transfer the ideas to a variety of topics.14 It’s as if they don’t want you to learn how to say a specific Albanian or Lithuanian or Icelandic phrase meaning I run but rather to understand the more general idea that there is a category of words called verbs, which you conjugate.
The challenge is that it’s often easier to pick up on a mathematical idea if it is applied directly to a concrete problem—even though that can make it more difficult to transfer the mathematical idea to new areas later. Unsurprisingly, there ends up being a constant tussle between concrete and abstract approaches to learning mathematics. Mathematicians try to hold the high ground by stepping back to make sure that abstract approaches are central to the learning process. In contrast, engineering, business, and many other professions all naturally gravitate toward math that focuses on their specific areas to help build student engagement and avoid the complaint of “When am I ever going to use this?” Concretely applied math also gets around the issue that many “real-world” word problems in mathematics textbooks are simply thinly disguised exercises. In the end, both concrete and abstract approaches have their advantages and disadvantages.
Transfer is beneficial in that it often makes learning easier for students as they advance in their studies of a discipline. As Professor Jason Dechant of the University of Pittsburgh says, “I always tell my students that they will study less as they progress through their nursing programs, and they don’t believe me. They’re actually doing more and more each semester; they just get better at bringing it all together.”
One of the most problematic aspects of procrastination—constantly interrupting your focus to check your phone messages, e-mails, or other updates—is that it interferes with transfer. Students who interrupt their work constantly not only don’t learn as deeply, but also aren’t able to transfer what little they do learn as easily to other topics.15 You may think you’re learning in between checking your phone messages, but in reality, your brain is not focusing long enough to form the solid neural chunks that are central to transferring ideas from one area to another.
TRANSFERRING IDEAS WORKS!
“I took fishing techniques from the Great Lakes and tried using them down in the Florida Keys this past year. Completely different fish, different bait, and a technique that had never been used but it worked great. People thought I was crazy and it was funny to show them that it actually caught fish.”
—Patrick Scoggin, senior, history
SUMMING IT UP
Equations are just ways of abstracting and simplifying concepts. This means that equations contain deeper meaning, similar to the depth of meaning found in poetry.
Your “mind’s eye” is important because it can help you stage plays and personalize what you are learning about.
Transfer is the ability to take what you learn in one context and apply it to something else.
It’s important to grasp the chunked essence of a mathematical concept, because then it’s easier to transfer and apply that idea in new and different ways.
Multitasking during the learning process means you don’t learn as deeply—this can inhibit your ability to transfer what you are learning.
PAUSE AND RECALL
Close the book and look away. What were the main ideas of this chapter? Can you picture some of these ideas with symbols in your mind’s eye?
ENHANCE YOUR LEARNING
1. Write an equation poem—several unfolding lines that provide a sense of what lies beneath a standard equation.
2. Write a paragraph that describes how some concepts you are studying could be visualized in a play. How do you think the actors in your play might realistically feel and react to one another?
3. Take a mathematical concept you have learned and look at a concrete example of how that concept is applied. Then step back and see if you can sense the abstract chunk of an idea underlying the application. Can you think of a completely different way that concept might be used?
INSIGHTS ON LEARNING FROM PHYSICS PROFESSOR BRAD ROTH, A FELLOW OF THE AMERICAN PHYSICAL SOCIETY AND CO-AUTHOR OF INTERMEDIATE PHYSICS FOR MEDICINE AND BIOLOGY
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Brad Roth and his dog Suki, enjoying the Michigan fall color.
“One thing I stress in my classes is to think before you calculate. I really hate the ‘plug and chug’ approach that many students use. Also, I find myself constantly reminding students that equations are NOT merely expressions you plug numbers into to get other numbers. Equations tell a story about how the physical world works. For me, the key to understanding an equation in physics is to see the underlying story. A qualitative understanding of an equation is more important than getting quantitatively correct numbers out of it.
“Here are a few more tips:
1. “Often, it takes way less time to check your work than to solve a problem. It is a pity to spend twenty minutes solving a problem and then get it wrong because you did not spend two minutes checking it.
2. “Units of measurement are your friend. If the units don’t match on each side of an equation, your equation is not correct. You can’t add something with units of seconds to something with units of meters. It’s like adding apples and rocks—nothing edible comes of it. You can look back at your work, and if you find the place where the units stop matching, you probably will find your mistake. I have been asked to review research papers that are submitted to professional journals that contain similar unit errors.
3. “You need to think about what the equation means, so that your math result and your intuition match. If they don’t match, then you have either a mistake in your math or a mistake in your intuition. Either way, you win by figuring out why the two don’t match.
4. “(Somewhat more advanced) For a complicated expression, take limiting cases where one variable or another goes to zero or infinity, and see if that helps you understand what the equation is saying.”
{ 1 }
open the door
What are the odds you’d open your refrigerator door and find a zombie in there, knitting socks? The odds are about the same that a touchy-feely, language-oriented person like me would end up as a professor of engineering.
Growing up, I hated math and science. I flunked my way through high school math and science courses, and only started studying trigonometry—remedial trigonometry—when I was twenty-six years old.
As a youngster, even the simple concept of reading a clock face didn’t seem to make sense to me. Why should the little hand point toward the hour? Shouldn’t it be the big hand, since the hour was more important than the minute? Did the clock read ten ten? Or one fifty? I was perpetually confused. Worse than my problems with clocks was the television. In those days before the remote control, I didn’t even know which button turned the television on. I watched a show only in the company of my brother or sister. They not only could turn the TV on, but could also tune the channel to the program we wanted to watch. Nice.
All I could conclude, looking at my technical ineptitude and flunking grades in math and science, was that I wasn’t very smart. At least, not that way. I didn’t realize it then, but my self-portrait as being technically, scientifically, and mathematically incapable was shaping my life. At the root of it all was my problem with mathematics. I had come to think of numbers and equations as akin to one of life’s deadly diseases—to be avoided at all costs. I didn’t realize then that there were simple mental tricks that could have brought math into focus for me, tricks that are helpful not only for people who are bad at math, but also for those who are already good at it. I didn’t understand that my type of thinking is typical of people who believe they can’t do math and science. Now, I realize that my problem was rooted in two distinctly different modes for viewing the world. Back then, I only knew how to tap one mode for learning—and the result was that I was deaf to the music of math.
Mathematics, as it’s generally taught in American school systems, can be a saintly mother of a subject. It climbs logically and majestically from addition through subtraction, multiplication, and division. Then it sweeps up toward the heavens of mathematical beauty. But math can also be a wicked stepmother. She is utterly unforgiving if you happen to miss any step of the logical sequence—and missing a step is easy to do. All you need is a disruptive family life, a burned-out teacher, or an unlucky extended bout with illness—even a week or two at a critical time can throw you off your game.
Or, as was the case with me, simply no interest or seeming talent whatsoever.
In seventh grade, disaster struck my family. My father lost his job after a serious back injury. We ended up in a hardscrabble school district where a crotchety math teacher made us sit for hours in the sweltering heat doing rote addition and multiplication. It didn’t help that Mr. Crotchety refused to provide any explanations. He seemed to enjoy seeing us flounder.
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Me at age ten with Earl the lamb. I loved critters, reading, and dreaming. Math and science weren’t on my play list.
By this time, I not only didn’t see any use for math—I actively loathed it. And as far as the sciences went—well, they didn’t. In my first chemistry experiment, my teacher chose to give my lab partner and me a different substance than the rest of the class. He ridiculed us when we fudged the data in an attempt to match everyone else’s results. When my well-meaning parents saw my failing grades and urged me to get help during the teacher’s office hours, I felt I knew better. Math and science were worthless, anyway. The Gods of Required Coursework were determined to shove math and science down my throat. My way of winning was to refuse to understand anything that was taught, and to belligerently flunk every test. There was no way to outmaneuver my strategy.
I did have other interests, though. I liked history, social studies, culture, and especially language. Luckily, those subjects kept my grades afloat.
Right out of high school, I enlisted in the army because they would actually pay me to learn another language. I did so well in studying Russian (a language I’d selected on a whim) that an ROTC scholarship came my way. I headed off to the University of Washington to get a bachelor’s degree in Slavic languages and literature, where I graduated with honors. Russian flowed like warm syrup—my accent was so good that I found myself on occasion mistakenly taken for a native speaker. I spent lots of time gaining this expertise—the better I got, the more I enjoyed what I was doing. And the more I enjoyed what I was doing, the more time I spent on it. My success reinforced my desire to practice, and that built more success.
But in the most unlikely situation I could have ever imagined, I eventually found myself commissioned as a second lieutenant in the U.S. Army Signal Corps. I was suddenly expected to become an expert in radio, cable, and telephone switching systems. What a turning point! I went from being on top of the world, an expert linguist, in control of my destiny, to being thrown into a new technological world where I was as stunted as a stump.
Yikes!
I was made to enroll in mathematically oriented electronics training (I finished at the bottom of the class), and then off I went to West Germany, where I became a pitiable communications platoon leader. I saw that the officers and enlisted members who were technically competent were in demand. They were problem solvers of the first order, and their work helped everyone accomplish the mission.
I reflected on the progress of my career and realized that I’d followed my inner passions without also being open to developing new ones. As a consequence, I’d inadvertently pigeonholed myself. If I stayed in the army, my poor technical know-how would always leave me a second-class citizen.
On the other hand, if I left the service, what could I do with a degree in Slavic languages and literature? There aren’t a lot of jobs for Russian linguists. Basically, I’d be competing for entry-level secretarial-type jobs with millions of others who also had bachelor’s of arts degrees. A purist might argue that I’d distinguished myself in both my studies and my service and could find much better work, but that purist would be unaware of how tough the job market can sometimes be.
Fortunately there was another unusual option. One of the great benefits of my service was that I had GI Bill money to offset the costs of future schooling. What if I used that support to do the unthinkable and try to retrain myself? Could I retool my brain from mathphobe to math lover? From technophobe to technogeek?
I’d never heard of anyone doing anything like that before, and certainly not coming from the phobic depths I’d sunk to. There couldn’t possibly be anything more foreign to my personality than mastering math and science. But my colleagues in the service had shown me the concrete benefits of doing so.
It became a challenge—an irresistible challenge.
I decided to retrain my brain.
It wasn’t easy. The first semesters were filled with frightening frustration. I felt like I was wearing a blindfold. The younger students around me mostly seemed to have a natural knack for seeing the solutions, while I was stumbling into walls.
But I began to catch on. Part of my original problem, I found, was that I had been putting my effort forth in the wrong way—like trying to lift a piece of lumber when you’re standing on it. I began to pick up little tricks about not only how to study but when to quit. I learned that internalizing certain concepts and techniques could be a powerful tool. I also learned not to take on too much at once, allowing myself plenty of time to practice even if it meant my classmates would sometimes graduate ahead of me because I wasn’t taking as many courses each semester as they were.
As I gradually learned how to learn math and science, things became easier. Surprisingly, just as with studying language, the better I got, the more I enjoyed what I was doing. This former Queen of the Confused in math went on to earn a bachelor’s degree in electrical engineering and then a master’s in electrical and computer engineering. Finally, I earned a doctorate in systems engineering, with a broad background that included thermodynamics, electromagnetics, acoustics, and physical chemistry. The higher I went, the better I did. By the time I reached my doctoral studies, I was breezing by with perfect grades. (Well, perhaps not quite breezing. Good grades still took work. But the work I needed to do was clear.)
Now as a professor of engineering, I have become interested in the inner workings of the brain. My interest grew naturally from the fact that engineering lies at the heart of the medical images that allow us to tease out how the brain functions. I can now more clearly see how and why I was able to change my brain. I also see how I can help you learn more effectively without the frustration and struggle I experienced.1 And as a researcher whose work straddles engineering, the social sciences, and the humanities, I’m also aware of the essential creativity underlying not just art and literature, but also math and science.
If you don’t (yet) consider yourself naturally good at math and science, you may be surprised to learn that the brain is designed to do extraordinary mental calculations. We do them every time we catch a ball, or rock our body to the beat of a song, or maneuver our car around a pothole in the road. We often do complex calculations, solving complex equations unconsciously, unaware that we sometimes already know the solution as we slowly work toward it.2 In fact, we all have a natural feel and flair for math and science. Basically, we just need to master the lingo and culture.
In writing this book, I connected with hundreds of the world’s leading professor-teachers of mathematics, physics, chemistry, biology, and engineering, as well as education, psychology, neuroscience, and professional disciplines such as business and the health sciences. It was startling to hear how often these world-class experts had used precisely the approaches outlined in the book when they themselves were learning their disciplines. These techniques were also what the experts asked their students to use—but since the methods sometimes seem counterintuitive, and even irrational, instructors have often found it hard to convey their simple essence. In fact, because some of these learning and teaching methods are derided by ordinary instructors, superstar teachers sometimes divulged their teaching and learning secrets to me with embarrassment, unaware that many other top instructors shared similar approaches. By collecting many of these rewarding insights in one place, you too can easily learn and apply practical techniques gleaned in part from these “best of the best” teachers and professors. These techniques are especially valuable for helping you learn more deeply and effectively in limited time frames. You’ll also gain insight from students and other fellow learners—people who share your constraints and considerations.
Remember, this is a book for math experts and mathphobes alike. This book was written to make it easier for you to learn math and science, regardless of your past grades in those subjects or how good or bad you think you are at them. It is designed to expose your thought processes so you can understand how your mind learns—and also how your mind sometimes fools you into believing you’re learning, when you’re actually not. The book also includes plenty of skill-building exercises that you can apply directly to your current studies. If you’re already good at numbers or science, the insights in this book can help make you better. They will broaden your enjoyment, creativity, and equation-solving elegance.
If you’re simply convinced you don’t have a knack for numbers or science, this book may change your mind. You may find it hard to believe, but there’s hope. When you follow these concrete tips based on how we actually learn, you’ll be amazed to see the changes within yourself, changes that can allow new passions to bloom.
What you discover will help you be more effective and creative, not only in math and science, but in almost everything you do.
Let’s begin!
references
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Amabile, TM, et al. “Creativity under the gun.” Harvard Business Review 80, 8 (2002): 52.
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index
The page numbers in this index refer to the printed version of this book. The link provided will take you to the beginning of that print page. You may need to scroll forward from that location to find the corresponding reference on your e-reader.
Page numbers in italics indicate photographs or illustrations.
abstractions, 197, 210, 212
See also chunking
abstractness and complexity, 16–17
Accounting Equation, 60
acne treatments, 128–29
acquaintances and success in job market, 231
activators for diffuse mode of thinking, 35
“active” repetition, chunking, 114, 119
active review, 142–43
“active” teaching technique, 218
addiction and procrastination, 87, 88–89
adding a new problem, chunking, 114
Afghanistan, 142
Africa, 215
“aha!” insights, 30, 227
See also diffuse mode of thinking
Alkon, Amy, 44
Allen, David, 126
alternating different problem-solving techniques, 257–58
Alzheimer’s, 45
anagrams exercise, 65
Anki flash cards, 64, 138, 174
anxiety, dealing with
choking vs. chunking, 112–25, 250–51
test taking, 103–4, 244–46, 248, 250–51
Appleyard, Nick, 192, 192
appreciating your talent, 183–92, 221
apps (best) for studying, 137–39
arsenic poison, 83, 89, 90, 145–46
attentional octopus, 14–15, 16, 52, 53, 53, 57
attention difficulties, 149, 150
Austen, Jane, 30
“autodidacts,” 222
Avogadro’s number, 163
avoidance and procrastination, 85–86
avoiding overconfidence, 20, 225–37, 247, 248, 254
background, working on a problem, 44
backward, working, 68
Baddeley, Alan, 62
“bad” trait, understanding value of, 221
ball bearing and Thomas Edison, 32
Bannister, Roger, 127
Batalha, Celso, 217–18
bed, bath, bus (three B’s), 30
befuddlement and learning, 22
Beilock, Sian, 103, 250, 250–51
belief part of zombies (habits), 88, 95, 99–100, 107
Bell, E. T., 223–24
Bellini, Marco, 206, 206
benzoyl peroxide and acne, 129
“big-picture” perspective, 12, 18, 19, 20, 30, 226, 227, 228, 247, 248
See also diffuse mode of thinking
big-picture top-down chunking process, 59, 60, 61, 61, 70, 79
Bilder, Robert, 49, 49–50
birds, survival instincts, 20
bite-sized pieces, breaking work into, 24–25, 97, 99, 103, 103–7, 104, 108, 132, 138, 149, 151, 151, 253, 258
Black-Derman-Toy model, 199
blinking and focusing, 37, 247, 248, 249
Blowers, Paul, 233
Bohr, Niels, 229–30, 230
bottom-up chunking process, 59, 60, 61, 61, 70, 79
Bradley, James V., xvii–xviii
Bradshaw, Bob, 242
brain
default settings of brain, 256
designed for extraordinary mental calculations, 6–7
maturity of brain, 195, 199
prefrontal cortex, 12, 12, 14–15, 187
procrastination and, 86–89, 87
retraining the brain, 5–6
sculpting your brain, 193–200
See also math and science, learning; neural structures, building
brainstorming, 229–33, 230
breaks, taking, 28, 30, 33, 34, 38, 47, 113, 114, 133, 134, 135, 258
breathing and test taking, 245, 248, 249
brick wall metaphor, 38, 38, 39, 43, 85, 254
Brisson, Charlene, 98
broad-perspective perceptual disorder of the right hemisphere, 226
Broadwell, Randall, 133
Buddha in Blue Jeans (Sheridan), 127
building a chunk, 56–61, 57, 60, 61, 78, 113–17
burnout, 145
Cajal, Santiago Ramón y, 193–94, 194, 195, 196–97, 200, 205, 206, 213, 215, 219, 221, 255
calculus limits, 169
Calculus Made Easy (Thompson), 169
Cameron, James, 216
cancer, 170
Carlsen, Magnus, 9–10, 10, 37, 185–86, 253
Carson, Ben, 214, 214
Cha, Mary, 142, 142–43
Chagnon, Napoleon, 223
change, possibility of, 88
changing habits, time for, 149
changing your thoughts and life, 195–96
checking your work, 228, 233, 236, 247, 248, 249
chemistry, 170, 171, 171, 176
chess, 9–10, 10, 36–37, 70, 71, 116–17, 146, 184, 185–86, 188, 253
Choke: What the Secrets of the Brain Reveal about Getting It Right When You Have To (Beilock), 103, 250
choking vs. chunking, 112–25, 250–51
chunking, 51–82, 112–25
adding a new problem for, 114
avoiding illusions of competence, 51–82, 256, 257
bottom-up chunking process, 59, 60, 61, 61, 70, 79
breaks, taking, 114
building a chunk, 56–61, 57, 60, 61, 78, 113–17
choking vs., 112–25, 250–51
context for, 58–59, 71
creativity and, 66–67, 67
deep chunking, 196–99, 198
defined, 54–55, 57, 57
focused attention, 52–54, 53, 54, 57, 78, 254, 255
illusions of competence, 61–68, 64, 67, 77, 79, 117, 125
interleaving vs. overlearning, 74, 74–78, 113, 173
knowledge collapse (hitting the wall), 118, 123
library of chunks, 66–68, 67, 113, 115, 117, 120, 121, 122, 147, 186
limited study time techniques, 81–82
memory traces, 53–54, 54, 58, 68, 69, 79, 185
mimicking solutions, avoiding, 77–78
neural structures, building, 52–53, 53, 54, 54–55, 67, 67, 68, 69, 71, 78, 93, 113, 121
organizing materials for, 73, 118–19
practice makes permanent, 68–72, 69, 74, 74, 78, 82, 120, 188, 257–58, 259
puzzle pieces metaphor, 61, 61, 74, 74
recall, 61–68, 64, 67, 72–73, 78, 78, 90, 116, 117, 123, 125
repetition of problem for, 114
summary, 78, 78, 121
testing effect, 119–20, 122, 238
top-down big-picture chunking process, 59, 60, 61, 61, 70, 79
transfer and, 59, 209–11, 212
understanding basic idea, 58, 78, 79
walking and recall, 30, 72–73, 90
working a problem through, 56–57, 58, 82, 114, 259
working memory and, 15, 41, 42, 42–43, 47, 64, 64, 65, 121, 122, 187
See also math and science, learning; memory
“chunk-puters,” 117
Cirillo, Francesco, 103
Click Moment, The (Johansson), 144–45
clock, problems reading, 1
Coffitivity, 139
coins and triangle exercise, 23, 23
“collaborative teaching” technique, 218
concept mapping, 71
concrete vs. abstract approach, 210, 212
context and chunking, 58–59, 71
continental drift example, 55
cortisol, 244
Coulton, Jonathan, 203
counterintuitive creativity, 19
Cowart, Aukury, 130
Coyne, Joseph, 105
CPR, 184
cramming, 24, 38, 38, 85, 87–88, 111, 145, 185, 254, 259
cranial bones mnemonic, 176
creative vs. nonimaginative scientists, 66
creativity and learning, 6, 29–50, 254–55
brick wall metaphor, 38, 38, 39, 43, 85, 254
chunking and, 66–67, 67
diffuse mode of thinking and, 32–33, 40
Einstellung effect (getting stuck), 17, 25, 26, 27–28, 36–39, 38, 52, 146, 170, 186, 243
failure, 33, 41, 110–11, 219
harnessing, extending abilities, 32–33
keeping up with the intellectual Joneses, 36
memory and, 179–80
neural structures, building, 32, 38, 38, 45, 46
summary, 46–47
talking with others for help, 40, 92, 260
toggling between thinking modes, 29–32, 31, 33–35, 36–39, 38, 46, 48
See also diffuse mode of thinking; math and science, learning; memory
Crick, Francis, xv–xvi
criticism, 50, 232
Crotchety, Mr., 3, 70
cue (trigger) part of zombies (habits), 88, 94, 95, 96, 107, 109, 129, 153
Dalí, Salvador, 31, 31, 32
Daniel, David B., 261–63
Darwin, Charles, 66, 208, 213
Day, Thomas, 63
day before a test, 246–47
daydreaming, 101, 187, 190
Dechant, Jason, 73, 177, 210–11
decimal system mnemonic, 176
deep-breathing and test taking, 245, 248, 249
deep chunking, 196–99, 198
default settings of brain, 256
Defense Language Institute, 53
“deliberate practice,” chunking, 114, 188
Dell, Michael, 216
density example, 173
depression, 45
Derman, Emanual, 199
developing the mind’s eye through equation poems. See mind’s eye, developing through equation poems
“Devil’s Advocate,” 228
See also right hemisphere of brain
Dickens, Charles, 30
different location and recall, 90, 182
different problems and techniques, interleaving, 75
diffuse mode of thinking, 11, 12
activators for, 35
“aha!” insights, 30, 227
background, working on a problem, 44
big-picture perspective, 12, 18, 19, 20, 30, 226, 227, 228, 247, 248
brick wall metaphor, 38, 38, 39, 43, 85, 254
chunk library and, 117
coins and triangle exercise, 23, 23
creativity and, 32–33, 40
flashlight metaphor, 18–19, 26
frustration signal, 39
hard-start–jump-to-easy technique, 241–44, 245–46, 248, 249
intuition problem solving, 67, 67, 236–37, 247
magical math marination, 142–43
pinball metaphor, 13, 13, 14, 15, 15
resting state network, 11
right hemisphere of brain, 20, 225–27, 227, 228
sentence errors exercise, 33
spirituality and, 189
triangles and squares exercise, 21, 21
See also creativity and learning; focused mode of thinking; toggling between thinking modes
disadvantages, turning into advantages, 193, 194, 195, 196, 197, 200
“disagreeableness” and creativity, 50
discomfort and procrastination, 85–86, 90, 101, 102, 107
distractions
memory and, 156, 260
procrastination and, 84, 105, 107, 150–51, 151, 154
double-checking your work, 228, 233, 236, 247, 248, 249
Dragone, Debra Gassner, 60
Drozd, Trevor, 38
Duhigg, Charles, 93
Earl (lamb), 3
easy does it, 9–28, 253
See also diffuse mode of thinking; focused mode of thinking
eBay, 124
Edison, Thomas, 29, 30–31, 31, 32, 33, 111
Einstein, Albert, 189, 206, 206, 230, 230
Einstellung effect (getting stuck), 17, 25, 26, 27–28, 36–39, 38, 52, 146, 170, 186, 243
electricity, 168
electromagnetic waves, 202
Ellison, Larry, 216
Emmett, Rita, 85
emotional tone-deafness, 226
empathy, 220
encryptedness and complexity, 16–17
energy savings from habits, 93–94
engines, invention, 112
enhancing your memory. See memory
epidermis layers, 162
equation poems, 201–5
See also mind’s eye, developing through equation poems
“equation sheet bingo,” 229
“Equation Whisperer.” See chunking
Evernote, 138
evolution, 213
exercise and memory, 108, 178–79, 182
experimental notebook, 107
expertise from memory tricks, 179–81
expertise (increasing your) and reducing anxiety. See chunking
explanatory questioning (simplifying study material), 79, 207–8, 255, 258
failure and learning, 33, 41, 110–11, 219
FAQs, procrastination, 148–52
fear and creativity, 49–50
fears, facing your, 244–46, 248, 250–51
Felder, Richard, 239, 239–41
Feynman, Richard, 202, 207, 229–30, 232, 252–53, 255, 256
Fields Medal, 70
Fiore, Neil, 129
fishing techniques, 211
flash cards, 64, 138, 174, 257
flashlight metaphor, 18–19, 26
focused attention, 52–54, 53, 54, 57, 78, 254, 255
focused mode of thinking, 11–12, 12
attentional octopus, 14–15, 16, 52, 53, 53, 57
brick wall metaphor, 38, 38, 39, 43, 85, 254
chunk library and, 117
coins and triangle exercise, 23, 23
flashlight metaphor, 18–19, 26
highly attentive state network, 11
left hemisphere of brain and, 20, 226, 228, 233
overconfidence and, 20, 226, 228, 233
pinball metaphor, 13, 13–14, 15, 15
prefrontal cortex, 12, 12, 14–15, 187
procrastination and, 24
sentence errors exercise, 33
sequential problem solving, 67, 67
triangles and squares exercise, 21, 21
willpower and, 34
See also diffuse mode of thinking; toggling between thinking modes
Foer, Joshua, 155–57, 156, 163
fooling yourself, ease of, 148, 229, 234
Fortenberry, Norman, 92, 92
43 Things, 139
Freedom, 138
“frogs,” eating first, 131, 140, 153, 258
frustration, diffuse mode of thinking, 39
furosemide memory trick, 181
Gabora, Liane, 32
Galois, Évariste, 224
Gamache, Robert R., 72
Gashaj, Michael, 137
Gates, Bill, 66, 216
Gazzaniga, Michael, 228
generation (recalling) effect, chunking, 115
genetic transposition (“jumping genes”), 206
genius envy, 185–89, 190
GI Bill, 5
goals, setting, 136, 137, 141, 152
Golden Apple Award, 208
Goldman Sachs, 199
Google, 27, 127, 138
Gordon, Cassandra, 41
Granovetter, Mark, 231
Gray-Grant, Daphne, 131
group work, 120, 130, 231–33, 234, 235, 239, 240, 241, 255, 259
Gruber, Howard, 30
habits. See zombies
hand bones mnemonic, 176
hand writing. See writing by hand
hard-start–jump-to-easy technique, 241–44, 245–46, 248, 249
hard tasks, 114, 116, 122, 148–49
Hardy, G. H., 223
harnessing, extending abilities, 32–33
harnessing your zombies (habits), 84, 95–101, 97
Hasan, Yusra, 96
Hebert, Susan Sajna, 246
hidden meanings in equations, 203–5, 211, 212
highlighting text, 62, 125, 178, 259
highly attentive state network, 11
hitting the wall (knowledge collapse), chunking, 118, 123
homework and test preparation, 240
hostage negotiation, 147
illusions of competence, 61–68, 64, 67, 77, 79, 117, 125
“impostor phenomenon,” 188
index cards example, 75–76
Inspire! program, 124
intellectual snipers caution, 219–21, 222
intention to learn and learning, 62
interleaving vs. overlearning, 74, 74–78, 113, 173
Intermediate Physics for Medicine and Biology (Roth), 236
internalizing concepts and solutions, 6, 73
introverts and teamwork, 233
intuition problem solving, 67, 67, 236–37, 247
intuitive understanding, 183–85, 185, 190
inventions, enhancing, 112, 113
Iraq, 80
isolation, 126, 130, 139, 153
James, William, 119
Jeshurun, Weston, 150
jingles, 163
Jobs, Steve, 216
Johansson, Frans, 144–45
Johnson, Steven, 66
Jordan, 168
“jumping genes” (genetic transposition), 206
“just this one time” phenomenon, 135–36
juvenile delinquents, 193–94, 199
Kamkwamba, William, 215
Kanigel, Robert, 223
Karpicke, Jeffrey, xvii–xviii, 61–62
Kasparov, Garry, 9–10, 10, 37
keeping up with the intellectual Joneses, 36
keys to becoming an “Equation Whisperer.” See chunking
“keystone” bad habit, procrastination, 86
knowledge collapse (hitting the wall), chunking, 118, 123
knowledge vs. memory trick, 176
Koehler, William, 180–81
Kruchko, Paul, 80, 80–82
labels and confidence, 192
language-learning skills, 1, 4, 5, 6, 14, 16, 53, 63, 70, 118, 145, 198, 209, 210
Law of Serendipity, ix, 66, 116, 122, 137, 256
learned industriousness, 99
learning more effectively, 6–7
See also math and science, learning
learning on your own, 213–16, 214, 218, 221, 222
learning to appreciate your talent. See appreciating your talent
LeechBlock, 139
left hemisphere of brain, 20, 226, 228, 233
Leopold, Kenneth R., 22
library of chunks, 66–68, 67, 113, 115, 117, 120, 121, 122, 147, 186
Limited Time Study, 81–82
limiting cases and understanding an equation, 237
Lisa, Allen, 93
Lisi, Garret, 104, 104
lists, enlisting, 130–31, 131, 132, 133, 134, 137, 140, 141, 149, 152
little goes a long way, 89
long-term memory, 41, 43, 47, 64, 65, 69, 74–75, 157, 179
magical math marination, 142–43
See also diffuse mode of thinking
“magic of creativity,” 32
Magrann, Tracey, 162, 164, 244
Mandelbrot, Benoit, 203
“Mandelbrot Set” (Coulton), 203
Manhattan Project, 229, 252
Man Who Knew Infinity: A Life of the Genius Ramanujan (Kanigel), 223
Manzi, Mr., 201–2
marking the objective, 102
marking up text, 62, 125, 178, 259
Marx, Chico, 136
math and science, learning
appreciating your talent, 183–92, 221
challenges of math and science, 16–18
easy does it, 9–28, 253
mind’s eye, developing through equation poems, 201–12
overconfidence, avoiding, 20, 225–37, 247, 248, 254
paradoxes of learning, 41, 255
renaissance learning, 213–24
sculpting your brain, 193–200
test taking, 238–51
unlocking your potential, 252–60
See also chunking; creativity and learning; diffuse mode of thinking; focused mode of thinking; memory; procrastination
maturity of brain, 195, 199
McClintock, Barbara, 206–7, 207
McCormick, Jonathon, 136
meaningful groups, memory, 175–76, 181
meditation, 126–27
MeeTimer, 139
memory, 155–82
creativity and, 179–80
distractions, greatest enemy, 156, 260
exercise and, 108, 178–79, 182
expertise from memory tricks, 179–81
jingles, 163
long-term memory, 41, 43, 47, 64, 65, 69, 74–75, 157, 179
meaningful groups, 175–76, 181
memory palace technique, 160–63, 161, 164, 165
metaphors, 168–71, 171, 175, 181, 182, 255, 258
mnemonics, 160, 175–76
muscle memory, 173, 177, 178–79, 181, 182
names of people, 174
neural structures, building, 159, 161, 162, 170, 173, 179, 182
senses, invoking, 159, 163, 164
short-term memory, 43
songs, 162, 181, 257
spaced repetition, 43, 47, 69, 172, 172–74, 257
spatial memory, 157–58, 161, 166–67
stories and, 177, 181
summary, 164, 181
talking to yourself, 178, 181
visualization, 156, 171, 171
visual memory, 157–58, 158, 158–59, 159, 161, 164, 165
visual metaphors, 168–71, 171, 175, 181, 182, 255, 258
working memory, 15, 41, 42, 42–43, 47, 64, 64, 65, 119, 121, 122, 157, 179, 186, 187
writing by hand, 173, 177, 178, 181
See also chunking; creativity and learning; math and science, learning
memory palace technique, 160–63, 161, 164, 165
memory traces, 53–54, 54, 58, 68, 69, 79, 185
Men of Mathematics (Bell), 223–24
mental contrasting technique, 99–100, 108, 258
mentors, value of, 216–18, 221
“metabolic vampires,” 172, 172, 174
metaphors
memory, 168–71, 171, 175, 181, 182, 255, 258
sculpting your brain, 198–99, 258
See also specific metaphors
Microsoft, 144–45
microtasks, 148
mimicking solutions, avoiding, 77–78
mindfulness and test taking, 245
mind’s eye, developing through equation poems, 201–12
“Mind Well” initiative, UCLA, 49
mineral hardness scale, 160
mini-testing, chunking, 119–20
MIT, 134
Miyoshi, Dina, 178
mnemonics, 160, 175–76
“mobile” phone invention, 112
monkeys in a benzene ring, 171, 171
months (thirty-one days), 159, 159
mornings and recall, 165, 182
motivating technique (mental contrasting), 99–100, 108, 258
multiple-choice tests, 246
multitasking caution, 106, 108, 211, 212
muscle memory, 173, 177, 178–79, 181, 182
musicians, 116
myelin sheaths, 195
names of people, memory, 174
negative feelings, dealing with, 127, 139, 140
negative self-talk, 250
neural structures, building
chunking, 52–53, 53, 54, 54–55, 67, 67, 68, 69, 71, 78, 93, 113, 121
creativity and learning, 32, 38, 38, 45, 46
easy does it, 11, 14–15, 16, 18, 24
memory, 159, 161, 162, 170, 173, 179, 182
sculpting your brain, 195–96, 197, 198
See also brain
neurological cravings, 93, 98
New Habit, The (Fiore), 129
Newman, Forrest, 102, 157
Newport, Cal, 134
Newton’s second law, 158, 158, 159, 172, 204
New York Times, 223
nightly, creating to-do list, 130, 131, 152
Nobel Prize winners, 194, 202, 206, 207, 219, 252
Noble Savages (Chagnon), 223
Noesner, Gary, 147
Nolta, Kathleen, 208
“not my fault,” 151–52, 153, 154
Noui-Mehidi, Nadia, 27, 27–28
“Now You Try!” challenges, 11, 23, 23, 24–25, 33, 39, 41, 44, 65, 68, 101, 107, 121, 135, 141, 152, 162, 175, 181, 209, 221
numbers, associating with events, 175
Oakley, Barbara, xix–xx, 1–8, 3
See also math and science, learning
“obvious,” struggling with, 219–20, 221–22
occipital lobe, 52
“Oh, Them Golden Slippers,” 163
opiate receptors, 216
organization and chunking, 73, 118–19
Orrell, Mike, 100, 117
overconfidence, avoiding, 20, 225–37, 247, 248, 254
overlearning vs. interleaving, 74, 74–78, 113, 173
panic, 103–4, 244–46, 248, 250–51
paradoxes of learning, 41, 255
Pariseau, Michael, 130
passion, following your, 147–48, 154
passive repetition, 119
pause and recall, 26, 79, 90, 108, 122, 140, 153, 165, 182, 190, 199, 212, 222, 234, 249, 257, 260
pause and reflect (wise waiting), 146–48, 154
Pavri, Vera, 177
persistence vs. intelligence, 101, 213–14, 221, 255
personalizing study material, 205–7, 206, 207, 208, 211, 212, 214–15
Pert, Candace, 215–16
photon, 206, 206
“picture walking,” 11, 61, 61, 81, 113, 260
Pietro, William, 176
pinball metaphor, 13, 13–16, 15, 16–17, 18
planner-journal, 130–37, 131, 140, 149, 152
Plath, Sylvia, 201, 202
“playtime” planning, 132, 134, 140
Ploughman, Elizabeth, 128
poetry and mathematics, 202–5, 211
Pomodoro technique, 24–25, 97, 99, 103, 103–7, 104, 108, 132, 138, 149, 151, 151, 253, 258
Porter, Mark, 189
Power of Habit, The (Duhigg), 93
power of teamwork (avoiding overconfidence), 20, 225–37, 247, 248, 254
practice makes permanent, 68–72, 69, 74, 74, 78, 82, 120, 188, 257–58, 259
practice tests, 246, 250
prefrontal cortex, 12, 12, 14–15, 187
Prentis, Jeffrey, 203–4
Presidential Medal of Freedom, 214
priming your mental pump, 11, 61, 61, 81, 113, 260
prioritizing and procrastination, 146
problem solving, chunking, 62–63, 65, 71
process vs. product, 101–2, 104, 104, 106, 107, 109
procrastination, 83–92, 144–54
addiction and, 87, 88–89
attention difficulties, 149, 150
brain and, 86–89, 87
challenge of, 17, 24, 26
change, possibility of, 88
cramming, 24, 38, 38, 85, 87–88, 111, 145, 185, 254, 259
discomfort and, 85–86, 90, 101, 102, 107
distractions and, 84, 105, 107, 150–51, 151, 154
FAQs, 148–52
focused mode of thinking and, 24
“just this one time” phenomenon, 135–36
smart people and, 188
summary, 90, 139–40, 152–53
tools, tips, and tricks, 126–43
transfer and, 211
willpower, 84, 91, 95, 97–98, 107, 150
wise waiting (pause and reflect), 146–48, 154
zombies (habits), 93–111
“zone, the,” working in, 144–46, 154
See also math and science, learning; zombies (habits)
Procrastination Equation, The (Steel), 96
pseudoscience as science, 229
puzzle pieces metaphor, 61, 61, 74, 74
Pythagorean theorem, 32
qualitative understanding of an equation, 236–37
questions, asking, 217, 260
quirky test questions, 218, 221
quitting time, planning, 132, 134, 141
Quizlet.com, 138
Ramachandran, V. S., 228
Ramanujan, Srinivasa, 223
random-access memory (RAM), 43
Ranjan, Apara, 32
RateMyProfessors.com, 110
recall and chunking, 61–68, 64, 67, 72–73, 78, 78, 90, 116, 117, 123, 125
rechecking your work, 228, 233, 236, 247, 248, 249
Reddit.com, 208
redos and creativity, 50
reducing anxiety and increasing your expertise. See anxiety, dealing with
reframing your focus, 127, 139, 140
refrigerator invention, 112
rehearsal and working memory, 42–43
relativity, 206
renaissance learning, 213–24
repetition of problem, chunking, 114
rereading, xvii–xviii, 61–62, 65, 71, 116, 117, 123, 125
resting state network, 11
retelling study material, 79, 207–8, 255, 258
retraining the brain, 5–6
reviewing material, 48, 58
rewarding yourself, 98, 99, 101, 106, 107, 108, 140, 152, 153
reward part of zombies (habits), 88, 95, 97, 98–99, 107
rewriting notes. See writing by hand
“Reykjavík Rapid” in 2004, 10, 10
right hemisphere of brain, 20, 225–27, 227, 228
Roberts, Seth, 128–29
Rohrer, Doug, 76
Rosenthal, Mike, 77–78
ROTC, 4
rote memorization, 113, 254
See also memory
Roth, Brad, 236, 236–37
routine part of zombies (habits), 84, 88, 94, 95, 96–97, 97, 107, 109, 129
rule following caution, 184
safecracking (Feynman), 252–53, 256
“safe environment” for criticism, 232
Sandburg, Carl, 30
Saucedo, Oraldo “Buddy,” 110, 110–11
schedule, freedom of a, 133
“school dreams,” 169
Schwalbe, Paul, 46
Science, 70
Science Olympiad, 189
Sciuto, Anthony, 169
Scoggin, Patrick, 211
sculpting your brain, 193–200
second nature, making knowledge, 63
seeking good advice from peers and teachers, 92, 260
Sejnowski, Terrence J., xv–xvi
self-control, 40, 48, 101
self-experimentation, 128–30
senses and memory, 159, 163, 164
sensory cortex, 187
sentence errors exercise, 33
sequential problem solving, 67, 67
setbacks, procrastination, 153, 154
setting goals, 136, 137, 141, 152
Shereshevsky, Solomon, 51, 52, 54, 55, 58
Sheridan, Tai, 127
Sherrington, Sir Charles, 205–6
short-term memory, 43
simplifying study material (explanatory questioning), 79, 207–8, 255, 258
SkillsToolbox.com, 163
Skinner, B. F., 146
sleep, 32, 34, 39, 44–46, 47, 60, 114, 153, 169, 182, 241, 248, 249, 260
“slower” students, 219–20, 221–22
“slow hunch,” 66
smart people and procrastination, 188
Smith, David Eugene, 202
snacks and memory, 164
songs and memory, 162, 181, 257
Sorby, Sheryl, 166, 166–67
Soviet Union and journalists, 51
spaced repetition, 43, 47, 69, 172, 172–74, 257
spatial memory, 157–58, 161, 166–67
spirituality, diffuse mode of thinking, 189
sports and math and science, 183
Stalling for Time (Noesner), 147
starting, zombies (habits), 105, 137, 141, 143, 152
StayFocusd, 139
Steel, Piers, 96
StickK, 139
“sticky student” syndrome, 217
stories and memory, 177, 181
“Strength of Weak Ties, The” (Granovetter), 231
stress, 103–4, 244–46, 248, 250–51
stuck, getting (Einstellung effect), 17, 25, 26, 27–28, 36–39, 38, 52, 146, 170, 186, 243
StudyBlue, 138
study groups, 120, 130, 231–33, 234, 235, 239, 240, 241, 255, 259
studying
rules, 257–60
strategies, 6–8, 261–63
See also math and science, learning
success, desire to practice, more success, 4, 6, 86, 99
success vs. GRE scores, 187
Sundaresan, Neel, 124, 124–25
synapse, 206
Syria, 168
talent, appreciating your. See appreciating your talent
talking to yourself, memory, 178, 181
talking with others for help, 40, 92, 260
Tan, Fabian Hadipriono, 174
TBI (traumatic brain injury), 80–81
Teacher of the Year Award, 105, 187
teachers, value of, 92, 216–18, 221, 260
teamwork, 225–37, 247, 248, 254
technology tips, studying, 137–39
TED talk, 163
teenagers and impulsive behavior, 195
television, problems using, 1–2
Ten Rules of Bad Studying, 259–60
Ten Rules of Good Studying, 257–58
testing effect, chunking, 119–20, 122, 238
testing yourself, 257
Test Preparation Checklist, 239–41, 248, 249
test taking, 238–51
tetracycline and acne, 128–29
thalamus, 52
thinking modes. See diffuse mode of thinking; focused mode of thinking
30/30, 138
Thompson, Silvanus, 169
thoughts, visual as well as verbal, 202
three B’s (bed, bath, bus), 30
Thurston, William, 70
time to learn, giving yourself, 6, 36
Titanic (ship), 223
toggling between thinking modes
creativity and learning, 29–32, 31, 33–35, 36–39, 38, 46, 48
easy does it, 10, 20–23, 21, 22, 23, 25, 253
See also diffuse mode of thinking; focused mode of thinking
tools, tips, and tricks, 126–43
See also procrastination
top-down big-picture chunking process, 59, 60, 61, 61, 70, 79
transfer, 59, 209–11, 212
transition metals, 176
traumatic brain injury (TBI), 80–81
Treacher-Collins mutant, 105
triangles and squares exercise, 21, 21
trigger (cue) part of zombies (habits), 88, 94, 95, 96, 107, 109, 129, 153
trying too hard can sometimes be part of the problem. See easy does it
twenty-five-minute timer (Pomodoro technique), 24–25, 97, 99, 103, 103–7, 104, 108, 132, 138, 149, 151, 151, 253, 258
underestimating yourself, 189
underlining text, 62, 125, 178, 259
understanding basic idea for chunking, 58, 78, 79
units of measurement, friendly, 236, 247
unlocking your potential, 252–60
“upkeep” repetitions, 63
U.S. Army, 4–5, 142
U.S. Memory Championships, 156, 157
visualization, 156, 171, 171
visual memory, 157–58, 158, 158–59, 159, 161, 164, 165
visual metaphors, memory, 168–71, 171, 175, 181, 182, 255, 258
Wade, Nicholas, 223, 223–24
walking and recall, 30, 72–73, 90
Wassell, Shaun, 19
Wegener, Alfred, 55
Weierstrass, Karl, 204
weight training, 34–35
Where Good Ideas Come From (Johnson), 66
Whitehouse, Malcolm, 163
why vs. how, 183–84, 185, 185
Williamson, Alexander, 30
willpower, 34, 84, 91, 95, 97–98, 107, 150
wise waiting (pause and reflect), 146–48, 154
working a problem through, chunking, 56–57, 58, 82, 114, 259
working memory, 15, 41, 42, 42–43, 47, 64, 64, 65, 119, 121, 122, 157, 179, 186, 187
work in physics, 204–5
World Wide Web, 66
Wozniak, Steve, 216
writing about negative thoughts, 250
writing by hand
chunking and, 63, 76–77, 118, 125–26, 173, 257
memory and, 173, 177, 178, 181
Young, Scott, 208
YouTube, 27
Zettler, Bill, 105, 187
zombies (habits), 93–111
belief part of, 88, 95, 99–100, 107
cue (trigger) part of, 88, 94, 95, 96, 107, 109, 129, 153
energy savings from habits, 93–94
harnessing your, 84, 95–101, 97
marking the objective, 102
mental contrasting technique, 99–100, 108, 258
multitasking caution, 106, 108, 211, 212
neurological cravings, 93, 98
Pomodoro technique, 24–25, 97, 99, 103, 103–7, 104, 108, 132, 138, 149, 151, 151, 253, 258
process vs. product, focus, 101–2, 104, 104, 106, 107, 109
reward part of, 88, 95, 97, 98–99, 107
routine part of, 84, 88, 94, 95, 96–97, 97, 107, 109, 129
starting and, 105, 137, 141, 143, 152
summary, 107–8
zombie alliance (planner-journal), 130–37, 131, 140, 149, 152
See also procrastination
“zone, the,” working in, 144–46, 154
Zuckerberg, Mark, 216
foreword
Your brain has amazing abilities, but it did not come with an instruction manual. You’ll find that manual in A Mind for Numbers. Whether you’re a novice or an expert, you will find great new ways to improve your skills and techniques for learning, especially related to math and science.
Henri Poincaré was a nineteenth-century mathematician who once described how he cracked a difficult mathematical problem that he had been intensively working on for weeks without success. He took a vacation. As he was getting on a bus in the south of France, the answer to the problem suddenly came to him, unbidden, from a part of his brain that had continued to work on the problem while he was enjoying his vacation. He knew he had the right solution even though he did not write down the details until he later returned to Paris.
What worked for Poincaré can work for you too, as Barbara Oakley explains in this insightful book. Surprisingly, your brain can also work on a problem even while you are sleeping and are not aware of anything. But it does this only if you concentrate on trying to solve the problem before falling asleep. In the morning, as often as not, a fresh insight will pop to mind that can help you solve the problem. The intense effort before a vacation or falling asleep is important for priming your brain; otherwise it will work on some other problem. There is nothing special about math or science in this regard—your brain will work just as hard at solving social problems as on math and science problems, if that is what has been on your mind recently.
You will find many more insights and techniques about how to learn effectively in this fascinating and timely book, which looks at learning as an adventure rather than hard labor. You will see how you can fool yourself about whether you actually know the material; you will find ways to hold your focus and space out your practice; and you will learn to condense key ideas so you can hold them more easily in your mind. Master the simple, practical approaches outlined here and you will be able to learn more effectively and with less frustration. This wonderful guide will enrich both your learning and your life.
—Terrence J. Sejnowski, Francis Crick Professor, Salk Institute for Biological Studies
{ 9 }
procrastination zombie wrap-up
We’ve swept through a number of issues related to procrastination in these last few chapters. But here are a few final thoughts that can shed new insight into procrastination.
The Pluses and Minuses of Working Unrelentingly in “The Zone”
A chance meeting of two Microsoft techies at a Friday-night party in 1988 resulted in an exciting solution to a major software stumbling block that Microsoft had basically given up on. The pair left the party to give the idea a shot, firing up a computer and going through the problematic code line by line. Later that evening, it was clear that they were onto something. That something, as Frans Johansson describes in his fascinating book The Click Moment, turned the nearly abandoned software project into Windows 3.0, which helped turn Microsoft into the global technology titan it is today.1 There are times when inspiration seems to erupt from nowhere.
These kinds of rare creative breakthroughs—relaxed moments of insight followed by mentally strenuous, all-out, late-night labor—are very different from a typical day of studying math and science. It’s rather like sports: Every once in a while, you have a day of competition when you need to give everything you have under conditions of extraordinary stress. But you certainly wouldn’t train every single day under those kinds of conditions.
On days when you are super productive and keep working away long into the night, you may get a lot done—but in subsequent days, if you look at your planner-journal, you may note that you are less productive. People who make a habit of getting their work done in binges are much less productive overall than those who generally do their work in reasonable, limited stints.2 Staying in the zone too long will send you toward burnout.3
An impending deadline can ratchet up stress levels, moving you into a zone where the stress hormones can kick in and assist in thinking. But relying on adrenaline can be a dangerous game, because once stress goes too high, the ability to think clearly can disappear. More important, learning math and science for an upcoming examination is very different from finishing a written report by a given due date. This is because math and science demand the development of new neural scaffolds that are different from the social, pictorial, and language-oriented scaffolds that our brains have evolved to excel at. For many people, math- and science-related scaffolds develop slowly, alternating focused-mode and diffuse-mode thinking as the material is absorbed. Especially when it comes to learning math and science, the bingeing excuse, “I do my best work under deadlines,” is simply not true.4
Remember the arsenic eaters at the beginning of these chapters on procrastination? Back in the 1800s, when arsenic eating took hold in one tiny Austrian population, people ignored how harmful it was long-term, even if tolerance could be built up. It’s a little like not recognizing the dangers of procras